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Approximate Inferenceusing MCMC
9.520 Class 22
Ruslan SalakhutdinovBCS and CSAIL, MIT
1
Plan
1. Introduction/Notation.
2. Examples of successful Bayesian models.
3. Basic Sampling Algorithms.
4. Markov chains.
5. Markov chain Monte Carlo algorithms.
2
References/Acknowledgements
• Chris Bishop’s book: Pattern Recognition and Machine
Learning, chapter 11 (many figures are borrowed from this book).
• David MacKay’s book: Information Theory, Inference, and
Learning Algorithms, chapters 29-32.
• Radford Neals’s technical report on Probabilistic Inference Using
Markov Chain Monte Carlo Methods.
• Zoubin Ghahramani’s ICML tutorial on Bayesian Machine Learning:
http://www.gatsby.ucl.ac.uk/∼zoubin/ICML04-tutorial.html
• Ian Murray’s tutorial on Sampling Methods:
http://www.cs.toronto.edu/∼murray/teaching/
3
Basic Notation
P (x) probability of x
P (x|θ) conditional probability of x given θ
P (x, θ) joint probability of x and θ
Bayes Rule:
P (θ|x) =P (x|θ)P (θ)
P (x)
where
P (x) =
∫P (x, θ)dθ Marginalization
I will use probability distribution and probability density interchangeably. It should be obvious from the context.
4
Inference Problem
Given a dataset D = {x1, ..., xn}:
Bayes Rule:
P (θ|D) =P (D|θ)P (θ)
P (D)
P (D|θ) Likelihood function of θ
P (θ) Prior probability of θ
P (θ|D) Posterior distribution over θ
Computing posterior distribution is known as the inference problem.
But:
P (D) =
∫P (D, θ)dθ
This integral can be very high-dimensional and difficult to compute.
5
Prediction
P (θ|D) =P (D|θ)P (θ)
P (D)
P (D|θ) Likelihood function of θ
P (θ) Prior probability of θ
P (θ|D) Posterior distribution over θ
Prediction: Given D, computing conditional probability of x∗ requires
computing the following integral:
P (x∗|D) =
∫P (x∗|θ,D)P (θ|D)dθ
= EP (θ|D)[P (x∗|θ,D)]
which is sometimes called predictive distribution.
Computing predictive distribution requires posterior P (θ|D).
6
Model Selection
Compare model classes, e.g. M1 andM2. Need to compute posterior
probabilities given D:
P (M|D) =P (D|M)P (M)
P (D)
where
P (D|M) =
∫P (D|θ,M)P (θ,M)dθ
is known as the marginal likelihood or evidence.
7
Computational Challenges• Computing marginal likelihoods often requires computing very high-
dimensional integrals.
• Computing posterior distributions (and hence predictive
distributions) is often analytically intractable.
• In this class, we will concentrate on Markov Chain Monte Carlo
(MCMC) methods for performing approximate inference.
• First, let us look at some specific examples:
– Bayesian Probabilistic Matrix Factorization
– Bayesian Neural Networks
– Dirichlet Process Mixtures (last class)
8
Bayesian PMF
1234567...
1 2 3 4 5 6 7 ...
5 3 ? 1 ... 3 ? 4 ? 3 2 ...
~~R U
V
UserFeatures
FeaturesMovie
We have N users, M movies, and integer rating values from 1 to K.
Let rij be the rating of user i for movie j, and U ∈ RD×N , V ∈ RD×M
be latent user and movie feature matrices:
R ≈ U⊤V
Goal: Predict missing ratings.
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Bayesian PMF
UVj i
Rij
j=1,...,Mi=1,...,N
σ
ΘV UΘ
ααV U Probabilistic linear model with Gaussianobservation noise. Likelihood:
p(rij|ui, vj, σ2) = N (rij|u
⊤i vj, σ
2)
Gaussian Priors over parameters:
p(U |µU ,ΛU) =N∏
i=1
N (ui|µu, Σu),
p(V |µV , ΛV ) =M∏
i=1
N (vi|µv, Σv).
Conjugate Gaussian-inverse-Wishart priors on the user and movie
hyperparameters ΘU = {µu,Σu} and ΘV = {µv,Σv}.
Hierarchical Prior.
10
Bayesian PMF
Predictive distribution: Consider predicting a rating r∗ij for user i
and query movie j:
p(r∗ij|R) =
∫∫p(r∗ij|ui, vj)p(U, V,ΘU ,ΘV |R)︸ ︷︷ ︸
Posterior over parameters and hyperparameters
d{U, V }d{ΘU ,ΘV }
Exact evaluation of this predictive distribution is analytically
intractable.
Posterior distribution p(U, V,ΘU ,ΘV |R) is complicated and does not
have a closed form expression.
Need to approximate.
11
Bayesian Neural NetsRegression problem: Given a set of i.i.d observations X = {xn}Nn=1
with corresponding targets D = {tn}Nn=1.
x0
x1
xD
z0
z1
zM
y1
yK
w(1)MD w
(2)KM
w(2)10
hidden units
inputs outputs
Likelihood:
p(D|X,w) =N∏
n=1
N (tn|y(xn,w), β2)
The mean is given by the outputof the neural network:
yk(x,w) =M∑
j=0
w2kjσ
( D∑
i=0
w1jixi
)
where σ(x) is the sigmoid function.
Gaussian prior over the network parameters: p(w) = N (0, α2I).
12
Bayesian Neural Nets
x0
x1
xD
z0
z1
zM
y1
yK
w(1)MD w
(2)KM
w(2)10
hidden units
inputs outputs
Likelihood:
p(D|X,w) =N∏
n=1
N (tn|y(xn,w), β2)
Gaussian prior over parameters:
p(w) = N (0, α2I)
Posterior is analytically intractable:
p(w|D,X) =p(D|w,X)p(w)∫p(D|w,X)p(w)dw
Remark: Under certain conditions, Radford Neal (1994) showed, as the number of
hidden units go to infinity, a Gaussian prior over parameters results in a Gaussian
process prior for functions.
13
Undirected Modelsx is a binary random vector with xi ∈ {+1,−1}:
p(x) =1
Zexp
( ∑
(i,j)∈E
θijxixj +∑
i∈V
θixi
).
where Z is known as partition function:
Z =∑
x
exp( ∑
(i,j)∈E
θijxixj +∑
i∈V
θixi
).
If x is 100-dimensional, need to sum over 2100 terms.
The sum might decompose (e.g. junction tree). Otherwise we need
to approximate.
Remark: Compare to marginal likelihood.
14
Monte Carlo
p(z) f(z)
z
For most situations we will be interestedin evaluating the expectation:
E[f ] =
∫f(z)p(z)dz
We will use the following notation: p(z) = p(z)Z .
We can evaluate p(z) pointwise, but cannot evaluate Z.
• Posterior distribution: P (θ|D) = 1P (D)P (D|θ)P (θ)
• Markov random fields: P (z) = 1Z exp(−E(z))
15
Simple Monte Carlo
General Idea: Draw independent samples {z1, ..., zn} from
distribution p(z) to approximate expectation:
E[f ] =
∫f(z)p(z)dz ≈
1
N
N∑
n=1
f(zn) = f
Note that E[f ] = E[f ], so the estimator f has correct mean (unbiased).
The variance:
var[f ] =1
NE
[(f − E[f ])2
]
Remark: The accuracy of the estimator does not depend on
dimensionality of z.
16
Simple Monte Carlo
In general:∫
f(z)p(z)dz ≈1
N
N∑
n=1
f(zn), zn ∼ p(z)
Predictive distribution:
P (x∗|D) =
∫P (x∗|θ,D)P (θ|D)dθ
≈1
N
N∑
n=1
P (x∗|θn,D), θn ∼ p(θ|D)
Problem: It is hard to draw exact samples from p(z).
17
Basic Sampling Algorithm
How to generate samples from simple non-uniform distributions
assuming we can generate samples from uniform distribution.
p(y)
h(y)
y0
1
Define: h(y) =∫ y
−∞p(y)dy
Sample: z ∼ U [0, 1].
Then: y = h−1(z) is a sample from p(y).
Problem: Computing cumulative h(y) is just as hard!
18
Rejection Sampling
Sampling from target distribution p(z) = p(z)/Zp is difficult.
Suppose we have an easy-to-sample proposal distribution q(z), such
that kq(z) ≥ p(z), ∀z.
z0 z
u0
kq(z0)kq(z)
p(z)
Sample z0 from q(z).
Sample u0 from Uniform[0, kq(z0)]
The pair (z0, u0) has uniform distribution
under the curve of kq(z).
If u0 > p(z0), the sample is rejected.
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Rejection Sampling
z0 z
u0
kq(z0)kq(z)
p(z)
Probability that a sample is accepted is:
p(accept) =
∫p(z)
kq(z)q(z)dz
=1
k
∫p(z)dz
The fraction of accepted samples depends on the ratio of the area
under p(z) and kq(z).
Hard to find appropriate q(z) with optimal k.
Useful technique in one or two dimensions. Typically applied as a
subroutine in more advanced algorithms.
20
Importance Sampling
Suppose we have an easy-to-sample proposal distribution q(z), such
that q(z) > 0 if p(z) > 0.
p(z) f(z)
z
q(z) E[f ] =
∫f(z)p(z)dz
=
∫f(z)
p(z)
q(z)q(z)dz
≈1
N
∑
n
p(zn)
q(zn)f(zn), zn ∼ q(z)
The quantities wn = p(zn)/q(zn) are known as importance weights.
Unlike rejection sampling, all samples are retained.
But wait: we cannot compute p(z), only p(z).
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Importance SamplingLet our proposal be of the form q(z) = q(z)/Zq:
E[f ] =
∫f(z)p(z)dz =
∫f(z)
p(z)
q(z)q(z)dz =
Zq
Zp
∫f(z)
p(z)
q(z)q(z)dz
≈Zq
Zp
1
N
∑
n
p(zn)
q(zn)f(zn) =
Zq
Zp
1
N
∑
n
wnf(zn), zn ∼ q(z)
But we can use the same importance weights to approximateZp
Zq:
Zp
Zq
=1
Zq
∫p(z)dz =
∫p(z)
q(z)q(z)dz ≈
1
N
∑
n
p(zn)
q(zn)=
1
N
∑
n
wn
Hence:
E[f ] ≈1
N
∑
n
wn
∑n wn
f(zn) Consistent but biased.
22
ProblemsIf our proposal distribution q(z) poorly matches our target distribution
p(z) then:
• Rejection Sampling: almost always rejects
• Importance Sampling: has large, possibly infinite, variance
(unreliable estimator).
For high-dimensional problems, finding good proposal distributions is
very hard. What can we do?
Markov Chain Monte Carlo.
23
Markov ChainsA first-order Markov chain: a series of random variables {z1, ..., zN}
such that the following conditional independence property holds for
n ∈ {z1, ..., zN−1}:
p(zn+1|z1, ..., zn) = p(zn+1|zn)
We can specify Markov chain:
• probability distribution for initial state p(z1).
• conditional probability for subsequent states in the form of transition
probabilities T (zn+1←zn) ≡ p(zn+1|zn).
Remark: T (zn+1←zn) is sometimes called a transition kernel.
24
Markov ChainsA marginal probability of a particular state can be computed as:
p(zn+1) =∑
zn
T (zn+1←zn)p(zn)
A distribution π(z) is said to be invariant or stationary with respect
to a Markov chain if each step in the chain leaves π(z) invariant:
π(z) =∑
z′
T (z←z′)π(z′)
A given Markov chain may have many stationary distributions. For
example: T (z←z′) = I{z = z′} is the identity transformation. Then
any distribution is invariant.
25
Detailed BalanceA sufficient (but not necessary) condition for ensuring that π(z) is
invariant is to choose a transition kernel that satisfies a detailed
balance property:
π(z′)T (z←z′) = π(z)T (z′←z)
A transition kernel that satisfies detailed balance will leave that
distribution invariant:
∑
z′
π(z′)T (z←z′) =∑
z′
π(z)T (z′←z)
= π(z)∑
z′
T (z′←z) = π(z)
A Markov chain that satisfies detailed balance is said to be reversible.
26
RecapWe want to sample from target distribution π(z) = π(z)/Z
(e.g. posterior distribution).
Obtaining independent samples is difficult.
• Set up a Markov chain with transition kernel T (z′←z) that leaves
our target distribution π(z) invariant.
• If the chain is ergodic, i.e. it is possible to go from every state to
any other state (not necessarily in one move), then the chain will
converge to this unique invariant distribution π(z).
• We obtain dependent samples drawn approximately from π(z) by
simulating a Markov chain for some time.
Ergodicity: There exists K, for any starting z, T K(z′←z) > 0 for all π(z′) > 0.
27
Metropolis-Hasting AlgorithmA Markov chain transition operator from current state z to a new
state z′ is defined as follows:
• A new ’candidate’ state z∗ is proposed according to some proposal
distribution q(z∗|z), e.g. N (z, σ2).
• A candidate state x∗ is accepted with probability:
min
(1,
π(z∗)
π(z)
q(z|z∗)
q(z∗|z)
)
• If accepted, set z′ = z∗. Otherwise z′ = z, or the next state is the
copy of the current state.
Note: no need to know normalizing constant Z.
28
Metropolis-Hasting AlgorithmWe can show that M-H transition kernel leaves π(z) invariant byshowing that it satisfies detailed balance:
π(z)T (z′←z) = π(z)q(z′|z)min
(1,
π(z′)
π(z)
q(z|z′)
q(z′|z)
)
= min (π(z)q(z′|z), π(z′)q(z|z′))
= π(z′) min
(π(z)
π(z′)
)q(z′|z)
q(z|z′), 1
)
= π(z′)T (z←z′)
Note that whether the chain is ergodic will depend on the particulars
of π and proposal distribution q.
29
Metropolis-Hasting Algorithm
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3
Using Metropolis algorithm to sample
from Gaussian distribution with
proposal q(z′|z) = N (z, 0.04).
accepted (green), rejected (red).
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Choice of Proposal
σmax
σmin
ρ
Proposal distribution:
q(z′|z) = N (z, ρ2).
ρ large - many rejections
ρ small - chain moves too slowly
The specific choice of proposal can greatly affect the performance of
the algorithm.
31
Gibbs SamplerConsider sampling from p(z1, ..., zN).
z1
z2
L
l
Initialize zi, i = 1, ..., N
For t=1,...,T
Sample zt+11 ∼ p(z1|z
t2, ..., z
tN)
Sample zt+12 ∼ p(z2|z
t+11 , xt
3, ..., ztN)
· · ·
Sample zt+1N ∼ p(zN |z
t+11 , ..., zt+1
N−1)
Gibbs sampler is a particular instance of M-H algorithm with proposals
p(zn|zi6=n) → accept with probability 1. Apply a series (component-
wise) of these operators.
32
Gibbs SamplerApplicability of the Gibbs sampler depends on how easy it is to sample
from conditional probabilities p(zn|zi6=n).
• For discrete random variables with a few discrete settings:
p(zn|zi6=n) =p(zn, zi6=n)∑zn
p(zn, zi6=n)
The sum can be computed analytically.
• For continuous random variables:
p(zn|zi6=n) =p(zn, zi6=n)∫
p(zn, zi6=n)dzn
The integral is univariate and is often analytically tractable or
amenable to standard sampling methods.
33
Bayesian PMFRemember predictive distribution?: Consider predicting a rating
r∗ij for user i and query movie j:
p(r∗ij|R) =
∫∫p(r∗ij|ui, vj)p(U, V,ΘU ,ΘV |R)︸ ︷︷ ︸
Posterior over parameters and hyperparameters
d{U, V }d{ΘU ,ΘV }
Use Monte Carlo approximation:
p(r∗ij|R) ≈1
N
N∑
n=1
p(r∗ij|u(n)i , v
(n)j ).
The samples (uni , vn
j ) are generated by running a Gibbs sampler, whose
stationary distribution is the posterior distribution of interest.
34
Bayesian PMFMonte Carlo approximation:
p(r∗ij|R) ≈1
N
N∑
n=1
p(r∗ij|u(n)i , v
(n)j ).
The conditional distributions over the user and movie feature vectors
are Gaussians → easy to sample from:
p(ui|R,V,ΘU , α) = N(ui|µ
∗i ,Σ
∗i
)
p(vj|R,U,ΘU , α) = N(vj|µ
∗j ,Σ
∗j
)
The conditional distributions over hyperparameters also have closed
form distributions → easy to sample from.
Netflix dataset – Bayesian PMF can handle over 100 million ratings.
35
MCMC: Main ProblemsMain problems of MCMC:
• Hard to diagnose convergence (burning in).
• Sampling from isolated modes.
More advanced MCMC methods for sampling in distributions with
isolated modes:
• Parallel tempering
• Simulated tempering
• Tempered transitions
Hamiltonian Monte Carlo methods (make use of gradient information).
Nested Sampling, Coupling from the Past, many others.
36
Deterministic Methods
• Laplace Approximation
• Bayesian Information Criterion (BIC)
• Variational Methods: Mean-Field, Loopy Belief Propagation along
with various adaptations.
• Expectation Propagation.
• · · ·
37